Stable space-like singularity formation for axi-symmetric and polarized near-Schwarzschild black hole interiors

TL;DR

This paper proves a stability result for the Schwarzschild black hole interior singularity under polarized axial symmetry perturbations, revealing complex Kasner-like asymptotics and introducing a novel wave-ODE approach to Einstein vacuum equations.

Contribution

It introduces a new method to analyze Einstein vacuum equations as coupled wave-ODE systems, demonstrating stability and detailed asymptotics of singularities in polarized axial symmetry.

Findings

Singularity persists under perturbations with axial symmetry.

Solution approaches a Kasner-type singularity with anisotropic behavior.

New wave-ODE framework for Einstein equations in symmetry class.

Abstract

We show a stability result for the Schwarzschild singularity (inside the black hole region) for the Einstein vacuum equations. The result is proven in the class of polarized axial symmetry, under perturbations of the Schwarzschild data induced on a hypersurface $\{r=\e\}$, $\e<<2M$. Our result is only partly a stability result, in that we show that while a (space-like) singularity persists under perturbations as above, the behaviour of the metric approaching the singularity is much more involved than for the Schwarzschild solution. Indeed, we find that the solution displays asymptocially-velocity-term-dominated dynamics and approaches a different Kasner solution at each point of the singularity. These Kasner-type asymptotics are very far from isotropic, since (as in Schwarzschild) there are two contracting directions and one expanding one. Our proof relies on energy methods and on a new approach to the EVE in axial symmetry, which we believe has wider applicability: In this symmetry class and under a suitable geodesic gauge, the EVE can be studied as a free wave coupled to (nonlinear) ODEs, which couple the geometry of the projected, 2+1 space-time to the free wave. The fact that the nonlinear part of the Einstein equations is described by ODEs lies at the heart of how one can overcome a certain linear instability exhibited by the singularity.